Properties

Label 15.1.551...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-5.515\times 10^{46}$
Root discriminant \(1306.48\)
Ramified primes $2,3,5,11,43$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184)
 
Copy content gp:K = bnfinit(y^15 - 185760*y^10 - 32099328*y^5 - 10208485085184, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184)
 

\( x^{15} - 185760x^{10} - 32099328x^{5} - 10208485085184 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $15$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(1, 7)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-55149257878625098649628587934867375000000000000\) \(\medspace = -\,2^{12}\cdot 3^{13}\cdot 5^{15}\cdot 11^{5}\cdot 43^{13}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(1306.48\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{4/5}3^{9/10}5^{23/20}11^{1/2}43^{9/10}\approx 2916.5309088754652$
Ramified primes:   \(2\), \(3\), \(5\), \(11\), \(43\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-7095}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{12}a^{3}$, $\frac{1}{24}a^{4}$, $\frac{1}{144}a^{5}$, $\frac{1}{144}a^{6}$, $\frac{1}{864}a^{7}-\frac{1}{6}a^{2}$, $\frac{1}{1728}a^{8}$, $\frac{1}{10368}a^{9}-\frac{1}{72}a^{4}$, $\frac{1}{95011335936}a^{10}-\frac{30611}{15344208}a^{5}-\frac{27743}{106557}$, $\frac{1}{95011335936}a^{11}-\frac{30611}{15344208}a^{6}-\frac{27743}{106557}a$, $\frac{1}{8170974890496}a^{12}+\frac{253541}{1319601888}a^{7}+\frac{677453}{3054634}a^{2}$, $\frac{1}{49025849342976}a^{13}+\frac{890429}{3958805664}a^{8}+\frac{252521}{27491706}a^{3}$, $\frac{1}{42\cdots 36}a^{14}-\frac{8910361}{680914574208}a^{9}-\frac{3160867}{197023893}a^{4}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{5}$, which has order $5$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{5}$, which has order $5$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{241}{5278407552}a^{10}-\frac{14117}{1704912}a^{5}-\frac{1471706}{35519}$, $\frac{11\cdots 15}{14639663345472}a^{14}-\frac{69\cdots 87}{1815772197888}a^{13}-\frac{17\cdots 53}{1361829148416}a^{12}-\frac{18\cdots 25}{439867296}a^{11}+\frac{14\cdots 29}{10556815104}a^{10}+\frac{46\cdots 45}{113485762368}a^{9}-\frac{26\cdots 45}{146622432}a^{8}-\frac{86\cdots 81}{146622432}a^{7}-\frac{43\cdots 25}{213114}a^{6}+\frac{26\cdots 81}{426228}a^{5}+\frac{23\cdots 53}{525397048}a^{4}-\frac{63\cdots 93}{3054634}a^{3}-\frac{11\cdots 01}{1527317}a^{2}-\frac{83\cdots 00}{35519}a+\frac{26\cdots 26}{35519}$, $\frac{75\cdots 15}{19519551127296}a^{14}-\frac{19\cdots 75}{453943049472}a^{13}+\frac{50\cdots 05}{170228643552}a^{12}+\frac{10\cdots 53}{959710464}a^{11}-\frac{10\cdots 45}{2639203776}a^{10}+\frac{10\cdots 45}{591071679}a^{9}-\frac{17\cdots 95}{879734592}a^{8}+\frac{42\cdots 63}{3054634}a^{7}+\frac{11\cdots 29}{232488}a^{6}-\frac{31\cdots 51}{1704912}a^{5}+\frac{13\cdots 60}{65674631}a^{4}-\frac{10\cdots 25}{4581951}a^{3}+\frac{24\cdots 71}{1527317}a^{2}+\frac{18\cdots 92}{3229}a-\frac{78\cdots 62}{35519}$, $\frac{10\cdots 31}{21\cdots 68}a^{14}+\frac{95\cdots 47}{49025849342976}a^{13}-\frac{42\cdots 13}{247605299712}a^{12}-\frac{63\cdots 81}{95011335936}a^{11}+\frac{64\cdots 83}{10556815104}a^{10}-\frac{66\cdots 49}{75657174912}a^{9}-\frac{30\cdots 85}{7917611328}a^{8}+\frac{18\cdots 97}{59981904}a^{7}+\frac{20\cdots 37}{15344208}a^{6}-\frac{23\cdots 99}{213114}a^{5}-\frac{56\cdots 33}{1182143358}a^{4}+\frac{21\cdots 31}{54983412}a^{3}+\frac{13\cdots 39}{833082}a^{2}-\frac{14\cdots 61}{106557}a-\frac{20\cdots 96}{35519}$, $\frac{65\cdots 67}{191646501977088}a^{14}-\frac{35\cdots 93}{190022671872}a^{13}-\frac{23\cdots 43}{4085487445248}a^{12}-\frac{50\cdots 59}{31670445312}a^{11}+\frac{19\cdots 95}{31670445312}a^{10}+\frac{98\cdots 11}{61901324928}a^{9}-\frac{17\cdots 57}{20458944}a^{8}-\frac{17\cdots 63}{659800944}a^{7}-\frac{95\cdots 65}{1278684}a^{6}+\frac{73\cdots 93}{2557368}a^{5}+\frac{40\cdots 13}{214935156}a^{4}-\frac{14\cdots 79}{142076}a^{3}-\frac{28\cdots 43}{9163902}a^{2}-\frac{31\cdots 61}{35519}a+\frac{12\cdots 18}{35519}$, $\frac{15\cdots 35}{383293003954176}a^{14}+\frac{21\cdots 41}{1361829148416}a^{13}-\frac{23\cdots 35}{9170566656}a^{12}-\frac{35\cdots 65}{47505667968}a^{11}-\frac{11\cdots 71}{2639203776}a^{10}-\frac{76\cdots 41}{10316887488}a^{9}-\frac{10\cdots 77}{36655608}a^{8}+\frac{35\cdots 47}{59981904}a^{7}+\frac{21\cdots 71}{15344208}a^{6}+\frac{67\cdots 61}{852456}a^{5}+\frac{24\cdots 17}{53733789}a^{4}-\frac{44\cdots 35}{18327804}a^{3}-\frac{15\cdots 43}{833082}a^{2}-\frac{49\cdots 29}{106557}a+\frac{12\cdots 95}{35519}$, $\frac{29\cdots 73}{14\cdots 12}a^{14}-\frac{72\cdots 19}{12256462335744}a^{13}-\frac{29\cdots 55}{4085487445248}a^{12}+\frac{51\cdots 57}{23752833984}a^{11}+\frac{10\cdots 61}{390993152}a^{10}-\frac{56\cdots 71}{14185720296}a^{9}+\frac{80\cdots 21}{7917611328}a^{8}+\frac{57\cdots 38}{41237559}a^{7}-\frac{56\cdots 37}{15344208}a^{6}-\frac{62\cdots 99}{1278684}a^{5}+\frac{49\cdots 47}{394047786}a^{4}+\frac{95\cdots 71}{54983412}a^{3}-\frac{70\cdots 35}{1527317}a^{2}-\frac{64\cdots 11}{106557}a+\frac{59\cdots 90}{35519}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 380972780086174900 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 380972780086174900 \cdot 5}{2\cdot\sqrt{55149257878625098649628587934867375000000000000}}\cr\approx \mathstrut & 3.13583493154811 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 185760*x^10 - 32099328*x^5 - 10208485085184); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_3\times F_5$ (as 15T11):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.1419.1, 5.1.13846144050000.12

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 30 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ R ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.3.0.1}{3} }^{5}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ R ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
2.2.5.8a1.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(3\) Copy content Toggle raw display 3.1.5.4a1.1$x^{5} + 3$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
3.1.10.9a1.2$x^{10} + 6$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(5\) Copy content Toggle raw display 5.3.5.15a4.1$x^{15} + 15 x^{13} + 15 x^{12} + 90 x^{11} + 180 x^{10} + 360 x^{9} + 810 x^{8} + 1215 x^{7} + 1890 x^{6} + 2673 x^{5} + 2835 x^{4} + 2840 x^{3} + 2430 x^{2} + 1230 x + 263$$5$$3$$15$$F_5\times C_3$$$[\frac{5}{4}]_{4}^{3}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.2$x^{2} + 22$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(43\) Copy content Toggle raw display 43.1.5.4a1.1$x^{5} + 43$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)